WASHINGTON — In a pair of unsigned opinions, the Supreme Court on Monday restored the rights of an adoptive mother who had split with her lesbian partner and reversed a murder conviction tainted by prosecutorial misconduct.

The adoption ruling reversed one by the Alabama Supreme Court, which had refused to recognize the woman’s adoptions of three children, which had been granted by a Georgia court in 2007.

The woman, identified in court papers as V.L., said she was overjoyed.

“I have been my children’s mother in every way for their whole lives,” she said in a statement. “I thought that adopting them meant that we would be able to be together always. When the Alabama court said my adoption was invalid and I wasn’t their mother, I didn’t think I could go on.”

The United States Supreme Court’s opinion, which had no noted dissents, said the Alabama court had violated the Constitution’s “full faith and credit” clause. “A state may not disregard the judgment of a sister state because it disagrees with the reasoning underlying the judgment or deems it to be wrong on the merits,” the opinion said.

The two women in the case, V.L. v. E.L., No. 15-648, were in a committed relationship that started in 1995 and lasted about 17 years. They shared a last name.

One of them, identified in court papers as E.L., gave birth to a child in 2002 and to twins in 2004, both times by insemination from an anonymous donor. They raised the children together in Alabama until they broke up in 2011, and the adoptive mother, V.L., continued to see the children for a time afterward.

When a dispute about the visits arose, V.L. turned to an Alabama court, which granted her visitation rights based on the Georgia adoption judgment. The Alabama Supreme Court reversed that, saying in an unsigned opinion that the Georgia judgment was not entitled to the “full faith and credit” ordinarily required by the Constitution “to the public acts, records and judicial proceedings of every other state.”

The Alabama Supreme Court reasoned that the Georgia court had misunderstood Georgia law in allowing the adoption, saying that “Georgia law makes no provision for a nonspouse to adopt a child without first terminating the parental rights of the current parents.”

In urging the Supreme Court to restore the visitation order while the justices consider whether to hear the case, lawyers for the adoptive mother said that Alabama was alone in refusing to recognize another state’s court judgment allowing the adoption of a child by a member of a same-sex couple.

In a second opinion on Monday, the Supreme Court reversed the murder conviction of Michael Wearry, who is on death row in Louisiana for the 1998 killing of Eric Walber, a youth who had been delivering pizzas. Prosecutors had no physical evidence tying Mr. Wearry to the crime, largely relying on testimony from two men whose accounts were open to doubt.

Prosecutors violated the Constitution by failing to disclose three kinds of information to the defense, the opinion said: police reports indicating that one witness had lied and had it in for Mr. Wearry, information that the second witness had sought to trade his testimony for a plea deal and medical records contradicting the prosecution’s version of events.

Justice Samuel A. Alito Jr., joined by Justice Clarence Thomas, dissented, saying the Supreme Court should have set down the case for full briefing and oral argument. “There is no question in my mind that the prosecution should have disclosed this information,” Justice Alito wrote, “but whether the information was sufficient to warrant reversing petitioner’s conviction is another matter.”

Monday’s orders returned some 40 cases to lower courts for further consideration in light of the Montgomery decision. Justices Alito and Thomas appended a boilerplate statement to each one, reciting various grounds on which courts could still rule against the affected prisoners.

The same two justices dissented from the court’s decision not to hear American Freedom Defense Initiative v. King County, No. 15-584, an appeal from a decision allowing Seattle’s public transit system to reject an advertisement showing 16 men under the words “Faces of Global Terrorism.” The ad, from a private advocacy group, was largely identical to one from the federal government, later withdrawn, that had been approved by the transit agency.

Federal appeals courts are divided over how the First Amendment applies to transit-system advertising.